Theorem fsumdivc

Description: A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006) (Revised by Mario Carneiro, 24-Apr-2014)

	Ref	Expression
Hypotheses	fsummulc2.1	|- ( ph -> A e. Fin )
	fsummulc2.2	|- ( ph -> C e. CC )
	fsummulc2.3	|- ( ( ph /\ k e. A ) -> B e. CC )
	fsumdivc.4	|- ( ph -> C =/= 0 )
Assertion	fsumdivc	|- ( ph -> ( sum_ k e. A B / C ) = sum_ k e. A ( B / C ) )

Proof

Step	Hyp	Ref	Expression
1		fsummulc2.1	|- ( ph -> A e. Fin )
2		fsummulc2.2	|- ( ph -> C e. CC )
3		fsummulc2.3	|- ( ( ph /\ k e. A ) -> B e. CC )
4		fsumdivc.4	|- ( ph -> C =/= 0 )
5	2 4	reccld	|- ( ph -> ( 1 / C ) e. CC )
6	1 5 3	fsummulc1	|- ( ph -> ( sum_ k e. A B x. ( 1 / C ) ) = sum_ k e. A ( B x. ( 1 / C ) ) )
7	1 3	fsumcl	|- ( ph -> sum_ k e. A B e. CC )
8	7 2 4	divrecd	|- ( ph -> ( sum_ k e. A B / C ) = ( sum_ k e. A B x. ( 1 / C ) ) )
9	2	adantr	|- ( ( ph /\ k e. A ) -> C e. CC )
10	4	adantr	|- ( ( ph /\ k e. A ) -> C =/= 0 )
11	3 9 10	divrecd	|- ( ( ph /\ k e. A ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
12	11	sumeq2dv	|- ( ph -> sum_ k e. A ( B / C ) = sum_ k e. A ( B x. ( 1 / C ) ) )
13	6 8 12	3eqtr4d	|- ( ph -> ( sum_ k e. A B / C ) = sum_ k e. A ( B / C ) )